I've noticed that the platter suspension on my 4000 seems a bit "tired".  By this I mean the platter is almost bottomed out with an LP on it...there's only about 1-2mm of give before it hits the bump stops.

Is it possible to adjust, or "tighten" the suspension at all?  Or does this indicate a need for replacement of the suspension springs?

You certainly can - there are three leaf springs which are just metals leaves - they are adjustable and easily reached - the service manual is on site and is a good one. However if you take off the top plate, all will become obvious. The top plate is released by pulling the wood surround forward. You can then remove all the aluminium plates. Be careful of the cartridge!

Is the platter level with the top plates when no record is present? Are the transit screws fully undone?

The springs should be fine there is a nut which allows adjustment of the platter height so check there first, well, after the service manual which explains the adjustment!

The platter sits below the plane of the top plate, even with the transport screws fully disengaged.  This seems wrong to me.  I'll have a look at the service manual to figure out how to adjust the springs. 

Many thanks for the suggestions!

Adjust the nut on top of the bearing so that there is 13 mm between the platter and the detector arm. This should be done with the transit screws locked.

Does anyone know the exact size of the spanner needed to adjust the suspension screws?  None of the tools in my collection fits the bill.

It will be a metric spanner, from 6 Mm and up in 1 Mm increases, below 6 Mm in 0.5 Mm